Solution to Briot and Bouquet problem on singularities of differential equations

HAL Id: hal-01707628

https://hal.archives-ouvertes.fr/hal-01707628v3

Preprint submitted on 6 Mar 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Solution to Briot and Bouquet problem on singularities of differential equations

Ricardo Pérez-Marco

To cite this version:

Ricardo Pérez-Marco. Solution to Briot and Bouquet problem on singularities of differential equations.

2018. �hal-01707628v3�

SINGULARITIES OF DIFFERENTIAL EQUATIONS

RICARDO P ´EREZ-MARCO

Abstract. We solve Briot and Bouquet problem on the existence of non-monodromic (multivalued) solutions for singularities of differential equations in the complex do- main. The solution is an application of hedgehog dynamics for indifferent irrational fixed points. We present an important simplification by only using a local hedgehog for which we give a simpler and direct construction of quasi-invariant curves which does not rely on complex renormalization.

1. Introduction.

We prove the following Theorem:

Theorem 1. Let f(z) = e^2πiαz + O(z²), α ∈ R −Q be a germ of holomorphic diffeomorphism with an indifferent irrational fixed point at 0.

There is no orbit of f distinct from the fixed point at 0 that converges to 0 by positive or negative iteration by f.

This Theorem solves the question of C. Briot and J.-C. Bouquet on singularities of differential equations from 1856 ([7]), as well as questions of H. Dulac (1904, [10], [11]), ´E. Picard (1896, [28]), P. Fatou (1919, [12]), and two more recent conjectures of M. Lyubich (1986, [16]).

The Theorem is trivial when the fixed point is linearizable, so, for the rest of the article, we assume that f is not linearizable.

The main difficulty is to understand the non-linearizable dynamics. The proof relies on hedgehogs and their dynamics discovered by the author in [24]. More precisely, we have from [24] the existence of hedgehogs:

2010Mathematics Subject Classification. 37 F 50, 37 F 25.

Key words and phrases. Complex dynamics, Holomorphic dynamics, indifferent fixed points, hedgehogs, analytic circle diffeomorphisms, rotation number, small divisors, centralizers.

.

1

Theorem 2 (Existence of hedgehogs). LetU be a Jordan neighborhood of 0such that f and f⁻¹ are defined and univalent on U, and continuous on U.¯

There exists a hedgehog K with the following properties:

• 0∈K ⊂U¯

• K is a full, compact and connected set.

• K∩∂U 6=∅.

• f(K) =f⁻¹(K) =K.

Moreover, f acts continuously on the space of prime-ends of C−K and defines an homeomorphism of the circle of prime-ends with rotation number α.

Figure 1. A hedgehog and its defining neighborhood.

In the proof we only need to consider local hedgehogs, i.e. a hedgehog associated to a small diskU =Dr0 withr₀ >0 small enough. LetK₀ be the hedgehog associated toDr0. The two following Theorems imply our main Theorem.

Theorem 3. Let (p_n/q_n)_n≥0 be the sequence of convergents of α. We have

n→+∞lim f_/K^±qn

0 = id_K0 , where the convergence is uniform onK₀.

Therefore all points of the hedgehog are uniformly recurrent, and no point on the hedgehog distinct from 0 converges to 0 by positive or negative iteration by f.

Theorem 4. Letz₀ ∈U−K₀ such that the positive, resp. negative, orbit(fⁿ(z₀))n≥0, resp. (f⁻ⁿ(z0))n≥0, accumulates a point on K0. Then this orbit accumulates all K0,

K₀ ⊂(fⁿ(z₀))n≥0 (resp.K₀ ⊂(f⁻ⁿ(z₀))n≥0) .

In particular this implies that if such an orbit (fⁿ(z0))n≥0 (resp. (f⁻ⁿ(z0))n≥0) accumulates 0∈ K₀ then it cannot converge to 0. Note that if f is not linearizable then it is clear that 0∈∂K₀. Indeed one can prove that the hedgehog K₀ has empty interior and K₀ = ∂K₀, but we don’t need to use this fact. We can just prove the previous Theorem for∂K₀.

The proof of these two Theorems are done by constructing quasi-invariant curves near the hedgehog. These are Jordan curves surrounding the hedgehog and almost invariant by high iterates of the dynamics. The quasi-invariance property is obtained for the Poincar´e metric of the complement of the hedgehog in the Riemann sphere.

Therefore, it is enough to carry out the construction for local hedgehogs, and for these we have a direct and simpler construction of quasi-invariant curves, that does not rely on complex renormalization techniques. Classical one real dimensional es- timates for smooth circle diffeomorphism combined with an hyperbolic version of Denjoy-Yoccoz Lemma in order to control the complex orbits for analytic circle dif- feomorphisms, are enough. This gives an important simplification for local hedgehogs of the proof of the main Theorem that was announced in [21].

2. Historical introduction on Briot and Bouquet problem.

In 1856 C. Briot and J.-C. Bouquet published a foundational article [7] on the local solutions of differential equations in the complex domain. They are particularly inter- ested in how a local solution determines uniquely the holomorphic function through analytic continuation. They consider a first order differential equation of a differential equation of the form

dy

dx =f(x, y) ,

wheref is a meromorphic function of the two complex variables (x, y)∈C² in a neigh- borhood of a point (x₀, y₀). A. Cauchy proved his fundamental Theorem on existence and uniqueness of local solutions¹: If f is finite and holomorphic in a neighborhood of (x0, y0) then there exists a unique holomorphic local solution y(x) satisfying the initial conditions

y(x₀) =y₀ .

In their terminology, Briot and Bouquet talk about “solutions monog`enes et mon- odromes”, “monog`ene” or monogenic meaningC-differentiable, i.e. holomorphic, and

“monodrome” or monodromic meaning univalued, since they also consider multival- ued solutions with non-trivial monodromy atx₀ ∈C.

1What is called today in Calculus books Cauchy-Lipschitz Theorem.

Briot and Bouquet start their article by giving a simple proof of Cauchy Theorem by the majorant series method. Then they consider the situation wheref is infinite or has a singularity at (x₀, y₀). They observe that even in Cauchy’s situation, we may get to such a point by a global analytic continuation of any solution. We assume for now on that (x₀, y₀) = (0,0). Writing down f as the quotient of two holomorphic germs

f(x, y) = A(x, y) B(x, y) ,

they study the situation whenA(0,0) =B(0,0) = 0 (they call these singularities “of the form ⁰₀”). This is done in Chapter III, starting in section 75 of [7]. After a simple change of variables, the equation reduces to

xdy

dx =ay+bx+O(2) ,

and a discussion starts considering the different cases for different values of the co- efficients a, b ∈ C. They prove the remarkable Theorem that if a is not a positive integer, then there always exists a holomorphic solutiony(x) in a neighborhood of 0 vanishing at 0 (Theorem XXVIII in section 80 of [7]). They show that this holomor- phic solution is the only monodromic one and in their proof of uniqueness (in section 81) the equation is put in the form

xdy

dx =y(a+O(2)) .

In this last form the holomorphic solution corresponds toy = 0.

After that they proceed to show that when the real part of a is positive there are infinitely many non-monodromic solutions (section 82 in [7]), i.e. holomorphic solutionsy(x) that are multivalued around 0∈C.

They make the claim in section 85 in [7] that when the real part of a is negative there are no other solutions, not even non-monodromic, other than the holomorphic solution found.

The proof of this statement contains a gap. Starting with the new form of the differential equation

xdy

dx =y(a+O_y(1)) +xyϕ(x, y) , they transform it into

dy

y + (A+By+. . .)dy=adt

t +ψ(x, y)dt ,

whereA+By+. . .is a holomorphic function of y near 0 and ψ is holomorphic near (0,0). Assuming by contradiction the existence of another solution, integration of the equation over a path from x₁ tox, y₁ =y(x₁), gives

log y

y₁

+ (A(y−y₁) +. . .) = log x

x₁ a

+ Z x

x1

ψ(x, y(x))dx .

They pretend that this is of the form log

y(x) y₁

= log x

x₁ a

+,

where is a small quantity, vanishing for x = x₁, and very small when x → 0, to get a contradiction using that for <a < 0, <log(x/x₁)^a → +∞ when x → 0 but

<log(y/y₁)→ −∞if y(x)→0.

Unfortunately is not small because since y(x) is not monodromic, the integral Z x

x1

ψ(x, y(x))dx

is not monodromic either, and if the path of integration spirals around 0 it can get arbitrarily large.

E. Picard observes ([28] Vol. II p.314 and p.317, 1893, see also Vol. III p.27 and´ 29, 1896) that with some implicit assumptions (that are not in [7]) the argument is correct if we approach x = 0 along a path of finite length where the argument of y(x) stays bounded or with a tangent at 0, trying (not very convincingly) to rebate L. Fuchs that pointed out the error in [13]. H. Poincar´e does not mention the error in his article [29] where he states Briot and Bouquet result without any restriction, and in his Thesis [30] where he studies the case where the real part of a is positive (and carefully avoids discussing further the other problematic case).

Picard, in his first edition of his “Trait´e d’Analyse” ([28], Vol. III, page 30, 1896), casts no doubt about the correction of Briot and Bouquet statement:

“Il resterait `a d´emontrer que ces deux int´egrales sont, en dehors de toute hypoth`ese, les seules qui passent par l’origine ou qui s’en rapprochent ind´efiniment. Je dois avouer que je ne poss`ede pas une d´emonstration rigoureuse de cette proposition, qui ne paraˆıt cependant pas douteuse.”²

2“It remains to prove that these two solutions are, without any assumption, the only ones passing through the origin or accumulating it. I have to admit that I don’t have a proof of this fact but it doesn’t seem doubtful.”

He refers to the two Briot-Bouquet holomorphic solutionsy(x) andx(y). His belief is probably reinforced by the saddle picture for real solutions that clearly only exhibit two real solutions in R² passing through the singularity.

A major progress came with the Thesis of H. Dulac published in 1904 in the Journal of the ´Ecole Polytechnique [10]. He proves the existence of an infinite number of distinct non-monodromic solutions whenais a negative rational number, thus proving than Briot and Bouquet original claim is always false in the rational situation. From the introduction of [10] we can read

“. . . on sait depuis bien longtemps, qu’il n’existe que deux courbes int´egrales r´eelles passant par l’origine. En est-il de mˆeme dans le champ complexe ? C’est une question qui restait en suspens et que les g´eom`etres penchaient `a trancher par l’affirmative (Picard, Trait´e d’Analyse, II (sic)³, p. 30). Or je prouve, au contraire, tout au moins dans le cas o`u α est rationnel, qu’il existe une infinit´e d’int´egrales y(x) s’annulant avec x (x tendant vers z´ero suivant une loi convenable) . . .” ⁴

After Dulac’s result Picard changed the quoted text in later editions of his Trait´e d’Analyse ([28], Vol. III, 3rd edition, p.30, 1928) into:

“On a longtemps pr´esum´e que ces int´egrales sont, en dehors de toute hypoth`ese, les seules qui passent par l’origine ou qui s’en rapprochent ind´efiniment. Dans un excellent travail sur les points singuliers des ´equations diff´erentielles, M. Dulac a d´emontr´e que la question ´etait tr`es complexe. Prenons, par exemple, l’´equation

xdy

dx +y(ν+. . .) = 0 ,

o`u ν est positif, ´equation `a laquelle peut toujours se ramener le cas o`u λ est n´egatif.

M. Dulac examine particuli`erement le cas o`uν est un nombre rationnelp/q, et montre qu’il y a alors, en g´en´eral, une infinit´e d’int´egrales pour lesquelles xet y tendent vers 0.” ⁵

3Volume III is the correct reference.

4“. . . from long time ago we know that there are only two real solutions passing through the origin. Is it the same in the complex? This is a question that remained open and that the geometers were inclined to decide in the affirmative (Picard, Trait´e d’Analyse, II (sic), p. 30). But, on the contrary, I prove, at least in the case when αis rational, that there are infinitely many solutions y(x) vanishing withx(xconverging to 0 under a suitable law). . .”

5“For a long time it was believed that, without any further condition, these are the only solutions passing through or accumulating the origin.

In an excellent work on the singular points of differential equations, M. Dulac has proved that the question is very complex. Take for instance the equation

xdy

dx+y(ν+. . .) = 0,

Dulac insisted in his Thesis that he had no answer for the irrational case ([10] p.4):

“1. ν est irrationnel. On a un col. H(x, y)existe formellement, mais est divergent, au moins dans certains cas. S’il y a des int´egrales pour lesquelles x et y tendent simultan´ement vers 0, et si l’on d´esigne parω et θ les arguments de x ety, quels que soient m et n, |x^myⁿω| et |x^myⁿθ| croissent ind´efiniment. Je ne puis me prononcer sur l’existence de ces int´egrales.”⁶

The expressionyx^νH(x, y) is a formal first integral of the solutions and he discuss its convergence in p.20. It is well known to Dulac that convergence of H solves the problem.

Then 30 years later he recalls that the problem remains unsolved ([11] p.31):

“Dans le cas2 (ν irrationnel, h(x, y) divergent), on ne sait s’il existe des solutions nulles autres que x= 0, y= 0. Ce sont l`a deux questions qu’il y aurait grand int´erˆet

`

a ´elucider.”⁷

Many results obtained by these distinguished geometers of the XIXth century where rediscovered in modern times, sometimes with a different point of view or language.

The original problem of singularities of differential equations of the form ⁰₀ (according to Briot and Bouquet terminology) is equivalent to study solutions of the holomorphic vector fieldX = (B, A) near (0,0)∈C²,

(x˙ =B(x, y)

˙

y =A(x, y)

The local geometry corresponds also to the study the holomorphic foliations onC² near the singular point (0,0) defined by the differential form

A(x, y)dx−B(x, y)dy= 0 .

whereν is positive, equation that we can always reduce the case whereλis negative.

M. Dulac examines specially the case whereνis a rational numberp/q, and proves that in general there are an infinite number of solutions for whichxandy converge to 0.”

6“1.ν is irrational. We have a saddle. H(x, y) exists formally, but is divergent, at least in certain cases. If there are solutions xand y which tend simultaneously to 0, and if we note ω and θ the arguments ofxandy, then for allmandn, |x^myⁿω| and|x^myⁿθ|must grow indefinitely. I cannot decide on the existence of such solutions.”

7“In case 2 (ν irrational,h(x, y) divergent), we don’t know if there are null solutions other than x= 0, y= 0.”

The situation of Briot and Bouquet problem corresponds to an irreducible singularity with a non-degenerate linear part,

(αy+O(2))dx+ (x+O(2))dy= 0 , where−α =a is Briot and Bouquet coefficient.

Whenα∈C−R+, andαis neither a negative integer nor the inverse of a negative integer, we are in the Poincar´e domain and the singularity is equivalent to the linear one. Whenαis a negative integer or its inverse, then we can conjugate the singularity to a finite Poincar´e-Dulac normal form (see [2] section 24). We assume α real and positive α > 0, which defines, in modern terminology, a singularity in the Siegel do- main. The singularity is formally linearizable, but the convergence of the linearization presents problems of Small Divisors. Precisely in this situation Dulac already proved in his Thesis the existence of non-linearizable singularities in section 12. This is a notable achievement that anticipates in several decades the non-linearization results for indifferent fixed points. The existence of Briot and Bouquet holomorphic solution proves the existence of two leaves of the holomorphic foliation crossing transversally at (0,0). This means that the singularity can be put into the form

αy(1 +O(2))dx+x(1 +O(2))dy= 0 .

Again y = 0 corresponds to the Briot and Bouquet holomorphic solution. It is now easy to make the link with the original Briot and Bouquet ⁰₀ singulatities of differential equations. Each solution y(x), distinct from the only monodromic solutiony(x) = 0, with initial data (x₀, y₀) close to (0,0), has a graph over the x-axes that corresponds to the leaf of the foliation passing through the point (x₀, y₀). The multivaluedness or non-monodromic character of the solution can be seen in the intersection of that leaf with a transversal {x = x₀}. The y-coordinates of these points of intersection give the different values taken by the non-monodromic solution that are obtained by following a path in the leaf that projects onto thex-axes into a path circling around x= 0.

The topology of the foliation is understood through a holonomy construction (see [17], and for the rational case see [9]): Taking a transversal{x=x₀} and lifting the circle C(0,|x₀|)⊂ {y= 0} in nearby leaves, the return map following this lift in the negative orientation, defines a germ of holomorphic diffeomorphism in one complex variable with a fixed point at (x₀,0) ⊂ {x = x₀}. Taking a local chart in this complex line, we have a local holomorphic diffeomorphism f ∈ Diff(C,0), f(0) = 0, and linearizing the equations we can compute its linear part at 0,

f(z) =e^2πiαz+O(z²) .

(to see this, note that yx^α is a first integral of the linearized differential form, thus is invariant of the solutions in the first order) Thus we get a germ of holomorphic

diffeomorphism with an indifferent irrational fixed point. It is obvious from the classical point of view that the local dynamics near 0 of this return map contains the information about the non-monodromic solutions starting at x = x₀. Thus we transform our original problem into a problem of holomorphic dynamics. Note that we can also reconstruct all the foliation and a neighborhood of (0,0)∈C² minus the leave{y = 0} by continuing the continuing the complex leaves from the transversal.

J.-F. Mattei and P. Moussu proved in [17] that two singularities in the Siegel domain with conjugated holonomies are indeed conjugated inC² by “pushing” the conjugacy along these leaves and using Riemann removability Theorem inC². J.-Ch. Yoccoz and the author proved in [27] that the set of dynamical conjugacy classes of holonomies is in bijection with the set of conjugacy classes of singularities in the Siegel domain.

The rational case was previously treated by J. Martinet and J.-P. Ramis ([18], [19]) by identifying the conjugacy invariants. This establish a full dictionary of the two problems. In particular, an interesting corollary is that Brjuno diophantine condition is optimal for analytic linearization of the singularity.

For our problem, the existence of non-monodromic solutions vanishing withxwhen x→0 following an appropriate path is equivalent to finding a leave that accumulates the singularity (0,0) but distinct from the Briot and Bouquet leaves {x = 0} and {y= 0}and a pathγ on this leave converging to (0,0). This pathγ projects properly in the{y= 0}plane into a spiral around (0,0) and converging to (0,0). The pathγ is homothopic in the leave to a path aboveC(0,|x₀|) such that the iterates of the return map converve to (x₀,0). Sinceπ₁(C^∗)≈Z, this gives an orbit of the return map that has a positive or negative orbit converging to the indifferent fixed point. Conversely, if we have such an orbit of the return map, we can push homothopicaly the path in the leave close to{x= 0}to make it converging to (0,0) (just using continuity of the foliation).

Proposition 5. When α ∈ R⁺ − Q, Briot and Bouquet non-existence of non- monodromic solutions vanishing at0 is equivalent to the existence of an orbit distinct from 0 that converges to 0 by iteration by the return map f or f⁻¹.

Since linearizable dynamics don’t have this property, we see that C-L. Siegel lin- earization theorem ([31], 1942) shows that Briot and Bouquet statement is true when α ∈ R+−Q satisfies the arithmetic linearization condition that was improved later by A.D. Brjuno ([8]) to the so called Brjuno’s condition

+∞

X

n=0

logq_n+1

q_n <+∞ .

The sequence of (pn/qn)n≥0 are the convergents of α. The positive answer to Briot and Bouquet question in the linearizable case, that corresponds to H(x, y) being convergent in Dulac’s notation, was already well known to Dulac in [10].

Indeed the non-existence of non-monodromic for singularities of differential equa- tions were well understood in the linearizable case, since H. Poincar´e [29] because linearization is equivalent to the existence of a first integral of the system of the form (see [10] section [])

I(x, y) = yx^αH(x, y).

Note also that to have non-monodromic solutions y(x) that accumulate into (but not converge to) 0 when x →0 is a simpler problem that is equivalent for the mon- odromy dynamics to have an orbit that accumulates 0 by positive or negative iteration.

This was solved in general in [24] by the discovery of hedgehogs and the result that almost all points in the hedgehog for the harmonic measure have a dense orbit in the hedgehog.

What remains to be elucidated for the Briot and Bouquet problem is the non- linearizable case, and more precisely the following problem:

Problem 6. Let α ∈ R−Q and f(z) = e^2πiαz +O(z²) be a germ of holomorphic diffeomorphism with an indifferent fixed point at0. Does there exists z₀ 6= 0 such that

n→+∞lim fⁿ(z₀) = 0 .

P. Fatou was confronted to this problem in his pioneer study of the dynamics of rational functions ([12], 1919) without knowing the relation to Briot and Bouquet problem. About fixed points (”points doubles” in Fatou’s terminology) of holomor- phic germs, which are indifferent, irrational and non-linearizable, Fatou writes [12]

p.220-221:

“Il reste `a ´etudier les points doubles dont le multiplicateur est de la forme e^iα, α

´etant un nombre r´eel inconmensurable avec π. Nous ne savons que fort peu de choses sur ces points doubles, dont l’´etude du point de vue qui nous occupe paraˆıt tr`es difficile.

(. . .) Existe-t-il alors des domaines dont les cons´equents tendent vers le point double

? Nous ne pouvons actuellement ni en donner d’exemple, ni prouver que la chose soit impossible . . .” ⁸

8“It remains to study fixed points with a multiplier of the forme^iα,αbeing a real number incom- mensurable withπ. We know little about these fixed points, and their study from our point of view appears very hard. (. . .) Are there any domain such that the positive iterates converge to the fixed point? We cannot give examples nor rule out this possibility.”

Fatou’s question is related to the question of the non-existence of wandering com- ponents of the Fatou set for rational functions. This was only proved in 1985 by D.

Sullivan [33]. Note that we do indeed have domains (that are not Fatou components) converging by iteration to rational indifferent fixed point as the local analysis of the rational case shows.

The non-existence of domains converging to an indifferent irrational fixed point was also conjectured by M. Lyubich in [16] p.73 (Conjecture 1.2), apparently unaware of Fatou’s question. Lyubich also conjectured (Conjecture 1.5 (a) [16] p.77) that for any indifferent irrational non-linearizable fixed point there is a critical orbit that converges to the fixed point.

The author proved in [24] the Moussu-Dulac Criterium : f is not linearizable if and only if f has an orbit accumulating the fixed point 0. We may think that this could give support to the existence of a converging orbit. The discovery of hedgehogs gave new tools for the understanding of the non-linearizable dynamics. Indeed, hedgehogs are the central tool in the final solution of all this problems:

Theorem 7. There is no orbit converging by positive or negative iteration to an indifferent irrational fixed point of an holomorphic map and distinct from the fixed point.

Therefore, the Briot and Bouquet problem has a positive solution in the irrational case. The questions of Dulac, Picard, Fatou are solved. Lyubich’s Conjecture 1.2 in [16] has a positive answer, but conjecture 1.5 (a) in [16] is false: For a generic rational function, there is no critical point converging to an indifferent irrational non- linearizable periodic orbits. There may be pre-periodic critical points to this orbit, but this is clearly non-generic. We may formulate a proper conjecture that has better chances to hold true:

Conjecture 8. Let f be a rational function of degree 2 or more, with an indifferent irrational non-linearizable fixed point z₀. There exists a critical point c₀ of f, such that

n→+∞lim 1 n

n−1

X

j=0

δ_f^j_(c0)→δz0 .

Theorem 7 was announced in [21] and a complete proof was given in the unpub- lished manuscript [25]. The proof given here concentrates on this particular Theorem and the solution of Briot and Bouquet problem, and not the many other properties of general hedgehog’s dynamics. The proof follows the same lines as in [25], but we have incorporated several new ideas that greatly improve and simplify the technical part of construction of quasi-invariant curves that are fundamental in the study of

the hedgehog dynamics. It was recently noticed in [26] an hyperbolic interpretation of Denjoy-Yoccoz Lemma that controls orbits of an analytic circle diffeomorphism g in a complex neighborhood of the circle. Then, when we control the non-linearity

||DlogDg||_C⁰ of g, we can construct directly the quasi-invariant curves without com- plex renormalization. The second observation if that in the proof of Theorem 7 we can work with local hedgehogs (small hedgehogs). Then the associated circle diffeo- morphism has a small non-linearity and the construction of quasi-invariant curves is easier.

3. Analytic circle diffeomorphisms.

3.1. Notations. We denote by T = R/Z the abstract circle, and S¹ = E(T) its embedding in the complex plane Cgiven by the exponential mapping E(x) = e^2πix.

We study analytic diffeomorphisms of the circle, but we prefer to work at the level of the universal covering, the real line, with its standard embedding R ⊂ C. We denote by D^ω(T) the space of non decreasing analytic diffeomorphisms g of the real line such that, for any x∈R, g(x+ 1) = g(x) + 1, which is the commutation to the generator of the deck transformationsT(x) =x+ 1. An element of the space D^ω(T) has a well defined rotation numberρ(g)∈R. The order preserving diffeomorphism g is conjugated to the rigid translationT_ρ(g):x7→x+ρ(g), by an orientation preserving homeomorphismh:R→R, such that h(x+ 1) =h(x) + 1.

For ∆ > 0, we note B_∆ = {z ∈ C;|=z| < ∆}, and A_∆ = E(B_∆). The subspace D^ω(T,∆) ⊂D^ω(T) is composed by the elements of D^ω(T) which extend analytically to a holomorphic diffeomorphism, denoted again byg, such thatgandg⁻¹ are defined on a neighborhood of ¯B_∆.

3.2. Real estimates. We refer to [36] for the results on this section. We assume that the orientation preserving circle diffeomorphism g is C³ and that the rotation numberα=ρ(g) is irrational. We consider the convergents (p_n/q_n)n≥0 ofα obtained by the continued fraction algorithm (see [14] for notations and basic properties of continued fractions).

For n ≥ 0, we define the map g_n(x) = g^qn(x) −p_n and the intervals I_n(x) = [x, g_n(x)],J_n(x) = I_n(x)∪I_n(g⁻¹_n (x)) = [g⁻¹_n (x), g_n(x)]. Letm_n(x) = g^qn(x)−x−p_n=

±|I_n(x)|, M_n = sup_R|m_n(x)|, and m_n = min_R|m_n(x)|. Topological linearization is equivalent to limn→+∞M_n = 0. This is always true for analytic diffeomorphisms by Denjoy’s Theorem, that holds for C¹ diffeomorphisms such that logDg has bounded variation.

Sinceg is topologically linearizable, combinatorics of the irrational translation (or the continued fration algorithm) shows:

Lemma 9. Let x ∈ R, 0 ≤ j < qn+1 and k ∈ Z the intervals g^j ◦T^k(In(x)) have disjoint interiors, and the intervals g^j◦T^k(J_n(x)) cover R at most twice.

We have the following estimates on the Schwarzian derivatives of the iterates off, for 0≤j ≤q_n+1,

Sg^j(x)

≤ M_ne^2VS

|In(x)|² ,

with S=||Sg||_C⁰_(R) and V = Var logDg.

This implies a control of the non-linearity of the iterates (Corollary 3.18 in [36]):

Proposition 10. For 0≤j ≤2q_n+1, c=√ 2Se^V,

||DlogDg^j||_C⁰_(R) ≤cMn^1/2

m_n .

These give estimates ong_n. More precisely we have (Corollary 3.20 in [36]):

Proposition 11. For some constant C > 0, we have

||logDg_n||_C⁰_(R) ≤CM_n^1/2 .

Corollary 12. For any >0, there exists n₀ ≥1 such that for n ≥n₀, we have

||Dg_n−1||_C⁰_(R)≤ .

Proof. Take n₀ ≥ 1 large enough so that for n ≥ n₀, CMn^1/2 < min(²₃,¹₂), then use Proposition 11 and |e^w−1| ≤ ³₂|w| for |w|<1/2.

Corollary 13. For any > 0, there exists n0 ≥ 1 such that for n ≥ n0, for any x∈R and y∈In(x) we have

1−≤ m_n(y)

m_n(x) ≤1 + . Proof. We have Dmn(x) =Dgn(x)−1, and

|m_n(y)−m_n(x)| ≤ ||Dm_n||_C⁰_(R)|y−x| ≤ ||Dg_n−1||_C⁰_(R)|m_n(x)| .

We conclude using Lemma 12.

4. Hyperbolic Denjoy-Yoccoz Lemma.

With these real estimates for the iterates, and, more precisely, a control on the non-linearity, we can use them to control orbits in a complex neighborhood. We give here a version of Denjoy-Yoccoz lemma (Proposition 4.4 in [36]) that is convenient for our purposes.

Given ∆ >0, we consider g ∈ D^ω(T,∆) such that inf_B∆<Dg > 0 so that logDg is a well defined univalued holomorphic function in B_∆. Given g ∈ D^ω(T) we get always this for a ∆ > 0 small enough (as in [36]), but here we don’t need to make the assumption that for a giveng, ∆ is small enough.

We do assume that we have a small non-linearity in B_∆, more precisely, τ =||DlogDg||_C⁰_(B∆) <1/9 .

Lemma 14. Let n₀ ≥1 large enough such that for all n ≥n₀, M_n<∆/2.

For x₀ ∈R, let 0< y₀ ≤1 and

z₀ =x₀+im_n(x₀)y₀ . Then for 0≤j ≤q_n+1, y_j ∈C, <y_j >0, is well defined by

z_j =g^j(z₀) = g^j(x₀) +im_n(g^j(x₀))y_j , and we have

|y_j−y₀| ≤ 3 4y₀ . Proof. For 0< t≤1 we define more generally

z_0,t=x₀+im_n(x₀)ty₀ , and we prove that y_j,t∈C, <y_j,t >0, is well defined by

zj,t =g^j(z0,t) =g^j(x0) +imn(g^j(x0))yj,t , and that we have

|yj,t−y0,t| ≤ 3 4y0,t .

Note that this last inequality implies <yj,t ≤ ⁷₄y0,t. The lemma corresponds to the case t= 1.

We prove this result by induction on 0 ≤ j < q_n+1 starting from j = 0 for which the result is obvious. Assuming it has been proved up to 0≤ j−1< q_n+1, then we have

0<=zj−1,t ≤M_n<yj−1,t ≤M_n7

4ty₀ < 7

8∆<∆,

so zj−1,t ∈B∆ and we can iterate once more and zj,t =g(zj−1,t) is well defined. We need to prove the estimate for y_j,t. By the chain rule we have

logDg^j(z0,t) =

j−1

X

l=0

logDg(zl,t). Therefore, we have

logDg^j(z0,t)−logDg^j(x0) ≤

j−1

X

l=0

|logDg(zl,t)−logDg(xl)|

≤τ

j−1

X

l=0

|z_l,t−x_l|

≤τ

j−1

X

l=0

|m_n(x_l)||y_l,t|

≤ 7 4τ ty₀

j−1

X

l=0

|m_n(x_l)|

≤ 7 4τ

j−1

X

l=0

|m_n(x_l)| .

Considering thej-iterate ofgon the intervalI_n(x₀), we obtain a pointζ ∈]x₀, g^qn(x₀)−

p_n[ such that,

Dg^j(ζ) = mn(xj) m_n(x₀) , and

logDg^j(ζ)−logDg^j(x₀)

≤τ|m_n(x₀)| ≤τ

j−1

X

l=0

|m_n(x_l)| . Adding the two previous inequalities, we have

logDg^j(z0,t)−logm_n(x_j) m_n(x₀)

≤ 11 4τ

j−1

X

l=0

|mn(xl)| . The intervalsI_n(x_l), 0≤l < q_n+1, being disjoint modulo 1, we have

qn+1−1

X

l=0

|m_n(x_l)|<1 .

So we obtain

logDg^j(z_0,T)−log m_n(x_j) m_n(x₀)

≤ 11 4 τ ,

and taking the exponential (using|e^w−1| ≤3/2|w|, for |w|<1/2, sinceτ < 1/9 and

11

4 τ < ¹₂), we have

Dg^j(z_0,t)− m_n(x_j) mn(x0)

≤ 33

8 τm_n(x_j) mn(x0). Now, integrating along the vertical segment [x₀, z_0,t] we get

g^j(z_0,t)−g^j(x₀)−iy₀m_n(x_j) ≤ 33

8 τ y_0,t|m_n(x_j)| , which, using τ <1/9, finally gives

|y_j,t−y_0,t| ≤ 11

24y_0,t < 3 4y_0,t .

4.1. Flow interpolation inR. Sinceg is analytic, from Denjoy’s Theorem we know that g_/R is topologically linearizable, i.e. there exists an non-decreasing homeomor- phismh:R→R, such that for x∈R, h(x+ 1) =h(x) + 1, and

h⁻¹◦g◦h=T_α , whereT_α :R→R, x7→x+α.

We can embed g into a topological flow on the real line (ϕ_t)_t∈R defined for t ∈ R byϕ_t =h◦T_tα◦h⁻¹. Wheng is analytically linearizable the diffeomorphisms of this flow are analytic circle diffeomorphisms, but in general, when g is not analytically linearizable the maps ϕ_t are only homeomorphism of the real line, although for t ∈ Z+α⁻¹Z,ϕ_tis analytic sinceϕ_tis an iterate ofg composed by an integer translation.

This can happen that for other values oft, whereϕ_tcan be an analytic diffeomorphism from the analytic centralizer of g since ϕ_t ◦g = g ◦ϕ_t. We refer to [22] for more information on this fact and examples of uncountable analytic centralizers for non- analytically linearizable dynamics. Now (ϕ_t)t∈[0,1] is an isotopy from the identity to g. The flow (ϕ_t)t∈R is a one parameter subgroup of homeomorphisms of the real line commuting to the translation by 1.

4.2. Flow interpolation in C. There are different complex extensions of the flow (ϕ_t)t∈R suitable for our purposes. For each n ≥ 0, we can extend this topological flow to a topological flow F_n in C by defining, for z₀ = x₀+i|m_n(x₀)|y₀ ∈ C, with x0, y0 ∈R,

ϕ⁽ⁿ⁾_t (z₀) = z₀(t) =ϕ_t(x₀) +i|m_n(ϕ_t(x₀))|y₀ .

We denote Φ⁽ⁿ⁾z0 the flow line passing through z₀, Φ⁽ⁿ⁾_z

0 = (ϕ⁽ⁿ⁾_t (z₀))t∈R.

4.3. Hyperbolic Denjoy-Yoccoz Lemma. We are now ready to give a geometric version of Denjoy-Yoccoz Lemma. We denote by d_P the Poincar´e distance in the upper half plane.

Lemma 15 (Hyperbolic Denjoy-Yoccoz Lemma). Let ∆>0 and g ∈D^ω(T,∆) such that

||D log Dg||_C⁰_(B∆)<1/9 . Let n₀ ≥1 large enough such that for all n ≥n₀, M_n<∆/2.

Let z₀ =x₀+i|m_n(x₀)|y₀, with 0< y₀ <1, so z₀ ∈B_∆. Then for 0≤j ≤q_n+1 we have that the (g^j(z₀)) piece of orbit follows at bounded distance the flow F_n for the Poincar´e metric of the upper half plane. More precisely we have

d_P(g^j(z₀), ϕ⁽ⁿ⁾_j (z₀))≤C₀ , for some constantC₀ >0 (we can take C₀ = 3).

Proof. We just use Lemma 14 reminding that the Poincar´e metric in the upper half plane is given by|ds|= ^|dξ|_=ξ and

d_P(z_j, ϕ⁽ⁿ⁾_j (z₀))≤ Z

[zj,ϕ⁽ⁿ⁾_j (z0)]

|dξ|

=ξ

≤ |m_n(x_j)|.|y_j −y₀| 1 inf_ξ∈[z

j,ϕ⁽ⁿ⁾_j (z0)]=ξ

≤ |m_n(x_j)|.|y_j −y₀| 4

|m_n(x_j)|y₀

≤4|y_j −y₀|

y₀ ≤3 =C₀

where in the second inequality we used that <y_j ≥ ¹₄y₀ which follow from |y_j−y₀| ≤

3

4y₀ that we also used in the last inequality.

5. Quasi-invariant curves for local hedgehogs.

Now we construct quasi-invariant curves for g under the previous assumptions:

g ∈D^ω(T,∆) and

τ =||DlogDg||_C⁰_(B∆) <1/9 .

Theorem 16 (Quasi-invariant curves). Let g be an analytic circle diffeomorphism with irrational rotation number α. Let (p_n/q_n)_n≥0 be the sequence of convergents of α given by the continued fraction algorithm.

Given C₀ >0 there is n₀ ≥0 large enough such that there is a sequence of Jordan curves (γn)n≥n0 for g which are homotopic to S¹ and exterior to D such that all the iterates g^j, 0 ≤ j ≤ qn, are defined in a neighborhood of the closure of the annulus U_n bounded by S¹ and γ_n, and we have

D_P(g^j(γ_n), γ_n)≤C₀ ,

where D_P denotes the Hausdorff distance between compact sets associated to d_P, the Poincar´e distance in C−D. We also have for any z ∈ γ_n, d_P(g^qn(z), z) ≤ C₀, that is,

||g^qn−id||_C^O

P(γn) ≤C₀ .

We choose the flow lines γn+1 = Φ⁽ⁿ⁾z0 , with y0 > 1/2 and n ≥ n0 for n0 ≥ 1 large enough, for the quasi-invariant curves of the Theorem. These flow lines are graphs overR. Given an interval I ⊂R, we label ˜I⁽ⁿ⁾ the piece of Φ⁽ⁿ⁾z0 overI.

Lemma 17. There is n0 ≥ 1 such that for n ≥ n0 and for any x ∈ R, the piece I˜n⁽ⁿ⁾(x) has bounded Poincar´e diameter.

Proof. Let z =x+i|mn(x)|y0 be the current point in ˜In⁽ⁿ⁾(x). We have dz = (1±i(Dg_n(x)−1)y₀) dx .

For any₀ >0, choosingn₀ ≥1 large enough, forn≥n₀, according to Lemma 12 we

have

dz dx −1

≤₀ . Therefore, we have

l_P( ˜I_n⁽ⁿ⁾(x₀)) = Z

I˜n⁽ⁿ⁾(x0)

1

|m_n(x)|y₀ |dz| ≤ Z

In(x0)

1

|m_n(x)|y₀ (1 +₀)dx . Now using Lemma 13 with =₀ and increasing n₀ if necessary, we have

l_P( ˜I_n⁽ⁿ⁾(x))≤ Z

In(x0)

1

|m_n(x₀)|y₀

1 +₀

1−₀ dx≤ 1 y₀

1 +₀

1−₀ ≤21 +₀

1−₀ ≤C .

We assumen ≥n₀ from now on in this section and the next one.

Lemma 18. For0≤j < q_n+1 and anyx∈R, the pieces (g^j◦T^k( ˜Jn⁽ⁿ⁾(x)))0≤j≤q_n+1,k∈Z

have bounded Poincar´e diameter and cover Φ⁽ⁿ⁾z0 .

Proof. From Lemma 17 any ˜In⁽ⁿ⁾(x) has bounded Poincar´e diameter, thus also any J˜n⁽ⁿ⁾(x) = ˜In⁽ⁿ⁾(x)∪I˜n⁽ⁿ⁾(g_n⁻¹(x)). Moreover, we haveg^j ◦T^k(J_n(x)) =J_n(g^j ◦T^k(x)), and all ˜Jn⁽ⁿ⁾(g^j ◦T^k(x)) have also bounded Poincar´e diameter. From Lemma 9 these

pieces cover Φ⁽ⁿ⁾z0 .

Corollary 19. For some C₀ >0, the flow orbit (ϕ⁽ⁿ⁾_j,k(z₀))0≤j<qn+1,k∈Z is C₀-dense in Φ⁽ⁿ⁾z0 for the Poincar´e metric.

We prove the first property stated in Theorem 16:

Proposition 20. Let γ_n = Φ⁽ⁿ⁻¹⁾z0 for some z₀ from the previous lemma, then we have, for 0≤j ≤q_n,

D_P(g^j(γ_n), γ_n)≤2C₀

Proof. We prove this Proposition for n + 1 instead of n (the proposition is stated to match n in Theorem 16). It follows from the hyperbolic Denjoy-Yoccoz Lemma that the orbit (g^j ◦T^k(z₀))0≤j<qn+1,k∈Z is C₀-close to flow orbit (ϕ⁽ⁿ⁾_j,k(z₀))0≤j<qn+1,k∈Z, and from Corollary 19 we have that a 2C₀-neighborhood of g^j(γ_n+1) contains γ_n+1. Conversely, since we can chooose any z₀ ∈ γ_n+1, we also have that g^j(γ_n+1) is in a

C₀-neighborhood of γ_n+1.

We prove the second property of Theorem 16. We observe thatg^qn+1(z₀)∈J˜n⁽ⁿ⁾(x₀), thatz₀ ∈J˜n⁽ⁿ⁾(x₀), and that ˜Jn⁽ⁿ⁾(x₀) has a bounded Poincar´e diameter by Lemma 18.

Thus we get (taking a larger C₀ >0 if necessary):

Proposition 21. For any z₀ ∈Φ⁽ⁿ⁾ , we have

d_P(z₀, g^qn+1(z₀))≤C₀ . 6. Osculating orbit.

We prove the existence of an osculating orbit.

Theorem 22 (Oscullating orbit). With the above hypothesis, forn ≥no there exists a quasi-invariant curves γ_n = Φ⁽ⁿ⁻¹⁾z0 such that the orbit (g^j(z₀))0≤j≤qn is such that the union of Poincar´e balls

U_n= [

0≤j<q_n,k∈Z

B_P(g^j(z₀) +k, C₀) ,

separatesR from {=z > H} withH >0large enough, and any orbit(g^j(w₀))j∈Z with

=w₀ > H with an iterate between γ_n and R has, for any 0≤j ≤q_n, an iterate in [

k∈Z

B_P(g^j(z₀) +k, C₀) .

From Lemma 18 we get the property that the hyperbolic balls B_P(ϕ⁽ⁿ⁾_t+k(z₀), C₀) cover Φ⁽ⁿ⁾z0 .

Lemma 23. We have that

U_n = [

0≤j<qn+1,k∈Z

B_P(ϕ⁽ⁿ⁾_t+k(z₀), C₀) is a neighborhood of the flow line Φ⁽ⁿ⁾z0

Proof. We prove Theorem 22. In the following argument C0 will denote several uni- versal constants. Enlarging the constantC₀, and using Lemma 15 we can replace the points ϕ⁽ⁿ⁾_t+k(z0) by the points g^j(z0) +k in the orbit of z0 in Lemma 23. Also, any orbit that jumps over γ_n (by positive or negative iteration) as in Theorem 22 has to visit aC0-neighborhood of γn , and will be C0-close to a point z1 ∈γn and then will beC0-close to theqn-orbit of z1 modulo 1. Finally we can replacez1 byz0 using that each point of the q_n-orbit of z₁ is C₀-close to a point in the q_n-orbit of z₀ modulo 1

(enlarge C₀ if need be).

7. Proof of the main Theorem.

We prove Theorems 3 and 4 that imply the main Theorem. We prove first the following preliminary Lemma that will allow us to work only with local hedgehogs.

Lemma 24. Let g_n ∈ D^ω(T,∆_n) with ρ(g_n) = α and ∆_n → +∞. Then g_n → R_α uniformly on compact sets of C^∗ and

n→+∞lim ||DlogDg_n||_C⁰_(R)= 0 .

Proof. Let ˜g_n be the associated circle diffeomorphism. The sequence (˜g_n) is a normal family in C^∗ (bounded inside D, and outside is the reflection across the unit circle), and any accumulation point is not constant since the unit circle is in the image of all

gn. Then by Hurwitz theorem any limit is an automorphism of C^∗, that extends to 0 by Riemann’s theorem, and so gives an automorphism of the plane leaving the unit circle invariant. The rotation number on the circle depends continuously on ˜g_n and is constant equal to α, therefore the only possible limit of the sequence (g_n) is R_α.

SinceDlogDR_α= 0 we get the last statement.

We consider now the hedgehogK₀ given by Theorem 2 for the domainU =Dr0, and we use the relation between hedgehogs and analytic circle diffeomorphisms presented in [24] to construct a circle diffeomorphism g₀.

Figure 2. Relation between hedgehogs and circle maps.

We consider a conformal representationh₀ :C−D→C−K₀ (D is the unit disk), and we conjugate the dynamics to a univalent map g₀ in an annulus V having the circleS¹ =∂D as the inner boundary,

g₀ =h⁻¹₀ ◦f◦h₀ :V →C .

The topology of K₀ is complex ([4], [5], [22]) and in particular K₀ is never locally connected, and h₀ does not extend to a continuous correspondence between S¹ and

∂K₀. Nevertheless, f extends continuously to Caratheodory’s prime-end compacti- fication of C−K0. This shows that g0 extends continuously to S¹ and its Schwarz reflection defines an analytic map of the circle defined on V ∪S¹ ∪V¯, where ¯V is the reflected annulus ofV. Then it is not difficult to see that g₀ is an analytic circle diffeomorphism. We can also prove thatg₀ has rotation number α. This is harder to prove in general (for an aribtrary hedgehog), but it is not difficult to show that we can pick K₀ so that the rotation number of g₀ is α (see [24] Lemma III.3.3) that is enough for our purposes. We choose such aK₀. Therefore, the dynamics in a complex neighborhood ofK₀ corresponds to the dynamics of an analytic circle diffeomorphism with rotation number α.

There is no risk of confusion and we denote alsog0 the lift toR. Theorem 25. Let ₀ >0 and ∆>0 be given

For r₀ >0 small enough we have g₀ ∈D^ω(T,∆) and

||DlogDg0||_C⁰_(R) < 0 .

Proof. Whenr₀ →0, we haveK₀ → {0}and the annulus whereg₀andg₀⁻¹are defined has a modulusM₀ →+∞. Therefore, by Gr¨otsch extremal problem, forr₀ >0 small enough we haveg₀ ∈D^ω(T,∆). From Lemma 24

rlim0→0||DlogDg₀||_C⁰_(R) = 0 ,

and the result follows.

Let₀ = 1/9 and ∆>0 be as in Section 5 and Section 6. We fix nowr₀ >0 small enough such that g₀ ∈D^ω(T,∆), ρ(g₀) =α, and

||DlogDg₀||_C⁰_(R) < ₀ ,

so that the hypothesis of Theorem 16 are fulfilled forg₀. Now we can apply Theorem 22 and find a sequence (γ_n)n≥n0 of quasi-invariant curves forg₀. We transport them byh₀ to get a sequence of Jordan curves (η_n)n≥n0

ηn =h0(γn) . We have

||g₀^qn−id||_C⁰

P(γn) ≤C₀ ,

therefore, for the Poincar´e metric of the exterior of the hedgehog,

||f^qn−id||_C⁰

P(ηn)≤C₀ , and, since ηn →K0, for the euclidean metric, we have

||f^qn−id||_C⁰_(ηn)=_n→0 .

Thus, if Ω_n is the Jordan domain bounded byη_n, by the maximum principle we have

||f^qn −id||_C⁰_(Ωn) =_n →0. Since Ω_n is a neighborhood of K₀, K₀ ⊂Ω¯_n, we have

||f^qn−id||_C⁰_(K0) =_n →0.

This proves Theorem 3 for the positive iterates (same proof for the negative ones, or just apply the result tof⁻¹).

We prove Theorem 4 for K₀, or more precisely for∂K₀ that was noted before that is enough for proving the Main Theorem (the hedgehog K₀ has empty interior and K₀ = ∂K₀, but we don’t need to use this fact). For the proof of Theorem 4 we